Annotations to a certain passage of Descartes for finding the quadrature of the circle
| dc.creator | Euler, Leonhard | |
| dc.date | 2007-05-23 | |
| dc.date | 2009-04-08 | |
| dc.date.accessioned | 2026-07-07T13:03:57Z | |
| dc.date.available | 2026-07-07T13:03:57Z | |
| dc.description | Translation from the Latin of "Annotationes in locum quendam Cartesii ad circuli quadraturam spectantem" (1763). The passage Euler is referring to is the "Excerpta" in part 6, p. 6 of Descartes' 1701 "Opuscula posthuma". Before reading this paper I had not heard of the "quadratrix" before, and I recommend learning a bit about it before reading this. I found Thomas Heath, "A history of Greek mathematics", vol. I, chapter VII to be helpful, in particular pp. 226-230. The quadratrix is a "mechanical curve" that can be used to rectify the circle. The usual problem of squaring the circle is to construct a square with the same area (or perimeter) as a given circle, in a finite number of steps using compass and straightedge. Descartes worked in the reverse direction: from a given square he constructed the radius of a circle with the same perimeter, but in an infinite number of steps. In this paper Euler reconstructs Descartes' argument and develops some consequences of it. Euler finds that \[ \sum_{n=0}^\infty \frac{1}{2^n} \tan \frac{1}{2^n}ϕ= \frac{1}ϕ - 2\cot 2ϕ. \] Integrating this yields \[ \prod_{n=1}^\infty \sec \frac{1}{2^n} ϕ= \frac{2ϕ}{\sin 2ϕ}. \] I'd like to thank Davide Crippa from the University of Paris 7 for some helpful back and forth about this paper. One of the only citations to this paper that I have found is in Pietro Ferroni, De calculo integralium exercitatio mathematica, Allegrini, Florence, 1792, pp. xxi--xxiii. The full text of it is available on Google Books. | |
| dc.description | 14 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/0705.3423 | |
| dc.identifier | http://arxiv.org/abs/0705.3423 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/226989 | |
| dc.subject | History and Overview | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 01A50; 51M15; 40A25 | |
| dc.title | Annotations to a certain passage of Descartes for finding the quadrature of the circle | |
| dc.type | text |